Phobos is a small moon in a circular orbit around Mars at an altitude of 6000 km above the surface of Mars. The gravitational field strength of Mars at its surface is 3.72 N kg\(^{-1}\). The radius of Mars is 3390 km. --- 5 WORK AREA LINES (style=lined) --- --- 7 WORK AREA LINES (style=lined) --- --- 5 WORK AREA LINES (style=lined) ---
PHYSICS, M5 2023 VCE 9*
An engineer is designing a banked circular curve of radius 25 m in a new bicycle velodrome.
Diagram A shows the bicycle approaching the banked section, and diagram B shows the front view of a bicycle moving out of the page as it rounds the banked bend.
The bicycle is travelling at 11 m s\(^{-1}\) on the banked section. At this speed there are no sideways frictional forces between the wheels and the road surface.
Determine the angle of the banked bend with the road surface, giving your answer to the nearest degree. (3 marks)
PHYSICS, M6 2023 VCE 5-6 MC
The diagram below shows a stationary circular coil of conducting wire connected to a low-resistance globe in a uniform, constant magnetic field, \(B\).
Question 5
The magnetic field is switched off.
Which one of the following best describes the globe in the circuit \( \textbf{before} \) the magnetic field is switched off, \( \textbf{during} \) the time the magnetic field is being switched off and \( \textbf{after} \) the magnetic field is switched off?
\begin{align*}
\begin{array}{l}
\rule{0pt}{2.5ex} \ \rule[-1ex]{0pt}{0pt}& \\
\rule{0pt}{2.5ex}\textbf{A.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex}\textbf{B.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex}\textbf{C.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex}\textbf{D.}\rule[-1ex]{0pt}{0pt}\\
\end{array}
\begin{array}{|c|c|}
\hline
\rule{0pt}{2.5ex}\quad\text{Before}\quad\rule[-1ex]{0pt}{0pt}&\quad \text{During} \quad& \quad\text{After}\quad\\
\hline
\rule{0pt}{2.5ex}\text{Off}\rule[-1ex]{0pt}{0pt}&\text{On}& \text{Off}\\
\hline
\rule{0pt}{2.5ex}\text{On}\rule[-1ex]{0pt}{0pt}& \text{On}& \text{Off}\\
\hline
\rule{0pt}{2.5ex}\text{On}\rule[-1ex]{0pt}{0pt}& \text{Off} & \text{Off}\\
\hline
\rule{0pt}{2.5ex}\text{Off}\rule[-1ex]{0pt}{0pt}& \text{On} & \text{On}\\
\hline
\end{array}
\end{align*}
Question 6
The radius of the coil is 5 cm and the magnetic field strength is 0.2 T. The coil has 100 loops. Assume that the magnetic field is perpendicular to the area of the coil.
Which one of the following is closest to the magnitude of the magnetic flux through the coil of wire when the magnetic field is switched on?
- 0.0016 Wb
- 0.16 Wb
- 16 Wb
- 1600 Wb
PHYSICS, M6 2023 VCE 1 MC
One type of loudspeaker consists of a current-carrying coil within a radial magnetic field, as shown in the diagram below. \(X\) and \(Y\) are magnetic poles, and the direction of the current, \(I\), in the coil is clockwise as shown.
The force, \(F\), acting on the current-carrying coil is directed into the page.
Which one of the following statements correctly identifies the magnetic polarities of \(X\) and \(Y\)?
- \(X\) is a north pole and \(Y\) is a south pole.
- \(X\) is a south pole and \(Y\) is a north pole.
- Both \(X\) and \(Y\) are north poles.
- Both \(X\) and \(Y\) are south poles.
Statistics, SPEC2 2023 VCAA 19 MC
A company accountant knows that the amount owed on any individual unpaid invoice is normally distributed with a mean of $800 and a standard deviation of $200.
What is the probability, correct to three decimal places, that in a random sample of 16 unpaid invoices the total amount owed is more than $13 500?
- 0.087
- 0.191
- 0.413
- 0.587
- 0.809
Probability, MET2 2023 VCAA 15 MC
Vectors, SPEC2 2023 VCAA 17 MC
Consider the vectors \(\underset{\sim}{\text{a}}=\alpha \underset{\sim}{\text{i}}+\underset{\sim}{\text{j}}-\underset{\sim}{\text{k}}, \ \underset{\sim}{\text{b}}=3 \underset{\sim}{\text{i}}+\beta \underset{\sim}{\text{j}}+4 \underset{\sim}{\text{k}}\) and \(\underset{\sim}{\text{c}}=2 \underset{\sim}{\text{i}}-7 \underset{\sim}{\text{j}}+\gamma \underset{\sim}{\text{k}}\), where \(\alpha, \beta, \gamma \in R\). If \(\underset{\sim}{\text{a}} \times \underset{\sim}{\text{b}}=\underset{\sim}{\text{c}}\), then
- \(\alpha=-2, \ \beta=-1, \ \gamma=-5\)
- \(\alpha=-1, \ \beta=2, \ \gamma=-1\)
- \(\alpha=1, \ \beta=-2, \ \gamma=-5\)
- \(\alpha=-2, \ \beta=-1, \ \gamma=-1\)
- \(\alpha=1, \ \beta=-2, \ \gamma=5\)
Mechanics, EXT2 M1 SM-Bank 5
A student throws a ball for his dog to retrieve. The position vector of the ball, relative to an origin \(O\) at ground level \(t\) seconds after release, is given by \( \underset{\sim}{\text{r}}{}_\text{B} (t)=5 t \underset{\sim}{\text{i}}+7 t \underset{\sim}{\text{j}}+(15 t-4.9 t^2+1.5) \underset{\sim}{\text{k}} \). Displacement components are measured in metres, where \(\underset{\sim}{\text{i}}\) is a unit vector to the east, \(\underset{\sim}{\text{j}}\) is a unit vector to the north and \( \underset{\sim} {\text{k}}\) is a unit vector vertically up.
Calculate the total vertical distance, in metres, travelled by the ball before it hits the ground. Give your answer correct to one decimal place. (3 marks)
Vectors, SPEC2 2023 VCAA 15 MC
If the sum of two unit vectors is a unit vector, then the magnitude of the difference of the two vectors is
- \(0\)
- \(\dfrac{1}{\sqrt{2}}\)
- \(\sqrt{2}\)
- \(\sqrt{3}\)
- \(\sqrt{5}\)
Calculus, SPEC2 2023 VCAA 13 MC
A tourist in a hot air balloon, which is rising vertically at 2.5 m s\(^{-1}\), accidentally drops a phone over the side when the phone is 80 metres above the ground.
Assuming air resistance is negligible, how long in seconds, correct to two decimal places, does it take for the phone to hit the ground?
- 2.86
- 2.98
- 3.79
- 4.04
- 4.30
Mechanics, EXT2 M1 EQ-Bank 6
The acceleration, \(a\) ms\(^{-2}\), of a particle that starts from rest and moves in a straight line is described by \(a=1+v\), where \(v\) ms\(^{-1}\) is its velocity after \(t\) seconds.
Determine the velocity of the particle after \( \log _e(e+1) \) seconds. (3 marks)
Probability, MET2 2023 VCAA 10 MC
A continuous random variable \(X\) has the following probability density function
\(g(x) = \begin {cases}
\dfrac{x-1}{20} &\ \ 1 \leq x < 6 \\
\\
\dfrac{9-x}{12} &\ \ 6 \leq x < 9 \\
\\ 0 &\ \ \ \text{elsewhere}
\end{cases}\)
The value of \(k\) such that \(\text{Pr}(X<k)=0.35\) is
- \(\sqrt{14}-1\)
- \(\sqrt{14}+1\)
- \(\sqrt{15}-1\)
- \(\sqrt{15}+1\)
- \(1-\sqrt{15}\)
Calculus, MET2 2023 VCAA 7 MC
Let \(f(x)=\log_{e}x\), where \(x>0\) and \(g(x)=\sqrt{1-x}\), where \(x<1\).
The domain of the derivative of \((f\circ g)(x)\) is
- \(x\in R\)
- \(x\in (-\infty, 1]\)
- \(x\in (-\infty, 1)\)
- \(x\in (0, \infty)\)
- \(x\in (0, 1)\)
Calculus, MET2 2023 VCAA 5 MC
Which one of the following functions has a horizontal tangent at \((0, 0)\)?
- \(y=x^{-\frac{1}{3}}\)
- \(y=x^{\frac{1}{3}}\)
- \(y=x^{\frac{2}{3}}\)
- \(y=x^{\frac{4}{3}}\)
- \(y=x^{\frac{3}{4}}\)
Algebra, MET2 2023 VCAA 4 MC
Consider the system of simultaneous equations below containing the parameter \(k\).
\(kx+5y\) | \(=k+5\) |
\(4x+(k+1)y\) | \(=0\) |
The value(s) of \(k\) for which the system of equations has infinite solutions are
- \(k\in \{-5, 4\}\)
- \(k\in \{-5\}\)
- \(k\in \{4\}\)
- \(k\in R\setminus \{-5, 4\}\)
- \(k\in R\setminus \{-5\}\)
Calculus, MET1 2022 VCAA 8
Part of the graph of `y=f(x)` is shown below. The rule `A(k)=k \ sin(k)` gives the area bounded by the graph of `f`, the horizontal axis and the line `x=k`.
- State the value of `A\left(\frac{\pi}{3}\right)`. (1 mark)
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- Evaluate `f\left(\frac{\pi}{3}\right)`. (2 marks)
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- Consider the average value of the function `f` over the interval `x \in[0, k]`, where `k \in[0,2]`.
- Find the value of `k` that results in the maximum average value. (2 marks)
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Calculus, MET1 2022 VCAA 7
A tilemaker wants to make square tiles of size 20 cm × 20 cm.
The front surface of the tiles is to be painted with two different colours that meet the following conditions:
- Condition 1 - Each colour covers half the front surface of a tile.
- Condition 2 - The tiles can be lined up in a single horizontal row so that the colours form a continuous pattern.
An example is shown below.
There are two types of tiles: Type A and Type B.
For Type A, the colours on the tiles are divided using the rule `f(x)=4 \sin \left(\frac{\pi x}{10}\right)+a`, where `a \in R`.
The corners of each tile have the coordinates (0,0), (20,0), (20,20) and (0,20), as shown below.
- i. Find the area of the front surface of each tile. (1 mark)
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ii. Find the value of `a` so that a Type A tile meets Condition 1. (1 mark)
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Type B tiles, an example of which is shown below, are divided using the rule `g(x)=-\frac{1}{100} x^3+\frac{3}{10} x^2-2 x+10`.
- Show that a Type B tile meets Condition 1. (3 marks)
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- Determine the endpoints of `f(x)` and `g(x)` on each tile. Hence, use these values to confirm that Type A and Type B tiles can be placed in any order to produce a continuous pattern in order to meet Condition 2. (2 marks)
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Graphs, MET1 2022 VCAA 6
The graph of `y=f(x)`, where `f:[0,2 \pi] \rightarrow R, f(x)=2 \sin(2x)-1`, is shown below.
- On the axes above, draw the graph of `y=g(x)`, where `g(x)` is the reflection of `f(x)` in the horizontal axis. (2 marks)
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- Find all values of `k` such that `f(k)=0` and `k \in[0,2 \pi]`. (3 marks)
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- Let `h: D \rightarrow R, h(x)=2 \sin(2x)-1`, where `h(x)` has the same rule as `f(x)` with a different domain.
- The graph of `y=h(x)` is translated `a` units in the positive horizontal direction and `b` units in the positive vertical direction so that it is mapped onto the graph of `y=g(x)`, where `a, b \in(0, \infty)`.
-
- Find the value for `b`. (1 mark)
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- Find the smallest positive value for `a`. (1 mark)
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- Hence, or otherwise, state the domain, `D`, of `h(x)`. (1 mark)
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- Find the value for `b`. (1 mark)
Probability, MET1 2022 VCAA 4
A card is drawn from a deck of red and blue cards. After verifying the colour, the card is replaced in the deck. This is performed four times.
Each card has a probability of `\frac{1}{2}` of being red and a probability of `\frac{1}{2}` of being blue.
The colour of any drawn card is independent of the colour of any other drawn card.
Let `X` be a random variable describing the number of blue cards drawn from the deck, in any order.
- Complete the table below by giving the probability of each outcome. (2 marks)
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- Given that the first card drawn is blue, find the probability that exactly two of the next three cards drawn will be red. (1 mark)
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- The deck is changed so that the probability of a card being red is `\frac{2}{3}` and the probability of a card being blue is `\frac{1}{3}`.
- Given that the first card drawn is blue, find the probability that exactly two of the next three cards drawn will be red. (2 marks)
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Calculus, SPEC2 2023 VCAA 11 MC
The area of the curved surface generated by revolving part of the curve with equation \(y=\cos ^{-1}(x)\) from \((0, \dfrac{\pi}{2})\) to \((1,0)\) about the \(y\)-axis can be found by evaluating
- \(2 \pi \displaystyle {\int_0^{\dfrac{\pi}{2}}\left(\cos ^{-1}(x) \sqrt{1+\dfrac{1}{x^2-1}}\right)} d x\)
- \(2 \pi \displaystyle {\int_0^1\left(\cos ^{-1}(x) \sqrt{1+\dfrac{1}{x^2-1}}\right)} d x\)
- \(2 \pi \displaystyle {\int_0^{\dfrac{\pi}{2}} \cos (y) \sqrt{1-\sin ^2(y)}} d y\)
- \(2 \pi \displaystyle {\int_0^{\dfrac{\pi}{2}} \sqrt{1+u^2} \ d u}\), where \(u=\sin (y)\)
- \(2 \pi \displaystyle {\int_0^1 \sqrt{1+u^2} \ d u } \), where \(u=\sin (y)\)
Calculus, SPEC2 2023 VCAA 10 MC
If \(I_n=\displaystyle {\int_0^1\left((1-x)^n e^x\right) d x}\), where \(n \in N\), then for \(n \geq 1, I_n\) equals
- \(-1+n I_{n-1}\)
- \(n I_{n-1}\)
- \(-1-n I_{n-1}\)
- \(-n I_{n-1}\)
- \((1-x)^n e^x+n I_{n-1}\)
Calculus, SPEC2 2023 VCAA 8 MC
Initially a spa pool is filled with 8000 litres of water that contains a quantity of dissolved chemical. It is discovered that too much chemical is contained in the spa pool water. To correct this situation, 20 litres of well-mixed spa pool water is pumped out every minute while 15 litres of fresh water is pumped in each minute.
Let \(Q\) be the number of kilograms of chemical that remains dissolved in the spa pool after \(t\) minutes. The differential equation relating \(Q\) to t is
- \(\dfrac{d Q}{d t}=\dfrac{4 Q}{t-1600}\)
- \(\dfrac{d Q}{d t}=\dfrac{-Q}{400}\)
- \(\dfrac{d Q}{d t}=\dfrac{3 Q}{t-1600}\)
- \(\dfrac{d Q}{d t}=\dfrac{3 Q}{1600-t}\)
- \(\dfrac{d Q}{d t}=\dfrac{4 Q}{1600-t}\)
Calculus, SPEC2 2023 VCAA 7 MC
Calculus, SPEC2 2023 VCAA 6 MC
Consider the following pseudocode.
define \(f(x, y)=e^{x y}\)
\(\begin{aligned} & x \leftarrow 0 \\ & y \leftarrow 0 \\ & h \leftarrow 0.5 \\ & n \leftarrow 0\end{aligned}\)
while \(n \geq 0\)
\(\begin{aligned} & y \leftarrow y+h \times f(x, y) \\ & x \leftarrow x+h \\ & n \leftarrow n+1\end{aligned}\)
print \(y\)
end while
After how many iterations will the pseudocode print 2.709 ?
- 1
- 2
- 3
- 4
- 5
Complex Numbers, SPEC2 2023 VCAA 5 MC
Let \(z\) be a complex number where \(\operatorname{Re}(z)>0\) and \(\operatorname{Im}(z)>0\).
Given \(|\bar{z}|=4\) and \(\arg \left(z^3\right)=-\pi\), then \(z^2\) is equivalent to
- \( {4z} \)
- \( -2 \bar{z} \)
- \( 3z \)
- \(\bar{z}^2\)
- \(-4 \bar{z}\)
Calculus, MET1 2023 VCAA 7
Consider \(f:(-\infty, 1]\rightarrow R, f(x)=x^2-2x\). Part of the graph of \(y=f(x)\) is shown below.
- State the range of \(f\). (1 mark)
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- Sketch the graph of the inverse function \(y=f^{-1}(x)\) on the axes above. Label any endpoints and axial intercepts with their coordinates. (2 marks)
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- Determine the equation of the domain for the inverse function \(f^{-1}\). (2 marks)
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- Calculate the area of the regions enclosed by the curves of \(f,\ f^{-1}\) and \(y=-x\). (2 marks)
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Calculus, MET1 2023 VCAA 5
- Evaluate \(\displaystyle \int_{0}^{\frac{\pi}{3}} \sin(x)\,dx\). (1 mark)
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- Hence, or otherwise, find all values of \(k\) such that \(\displaystyle \int_{0}^{\frac{\pi}{3}} \sin(x)\,dx=\displaystyle \int_{0}^{\frac{\pi}{2}} \cos(x)\,dx\), where \(-3\pi<k<2\pi\). (3 marks)
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Calculus, MET1 2023 VCAA 4
Calculus, MET1 2022 VCAA 1b
Find and simplify the rule of `f^{\prime}(x)`, where `f:R \rightarrow R, f(x)=\frac{\cos (x)}{e^x}`. (2 marks)
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Calculus, MET1 2022 VCAA 1a
Let `y=3xe^{2x}`.
Find `\frac{dy}{dx}`. (1 mark)
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Algebra, MET1 2023 VCAA 2
Solve \(e^{2x}-12=4e^{x}\) for \(x\ \in\ R\). (3 marks)
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Calculus, MET1 2023 VCAA 1a
Let \(y=\dfrac{x^2-x}{e^x}\).
Find and simplify \(\dfrac{dy}{dx}\). (2 marks)
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Trigonometry, SPEC2 2023 VCAA 3 MC
In the interval \(-\pi \leq x \leq \pi\), the graph of \(y=a+\sec (x)\), where \(a \in R\), has two \(x\)-intercepts when
- \(0 \leq a \leq 1\)
- \(-1<a<1\)
- \(a \leq-1\) or \(a>1\)
- \(-1 \leq a<0\)
- \(a<-1\) or \(a \geq 1\)
Vectors, SPEC1 2023 VCAA 10
The position vector of a particle at time \(t\) seconds is given by
\(\underset{\sim}{\text{r}}(t)=\big{(}5-6 \ \sin ^2(t) \big{)} \underset{\sim}{\text{i}}+(1+6 \ \sin (t) \cos (t)) \underset{\sim}{\text{j}}\), where \(t \geq 0\).
- Write \(5-6\, \sin ^2(t)\) in the form \(\alpha+\beta\, \cos (2 t)\), where \(\alpha, \beta \in Z^{+}\). (1 mark)
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- Show that the Cartesian equation of the path of the particle is \((x-2)^2+(y-1)^2=9.\) (2 marks)
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- The particle is at point \(A\) when \(t=0\) and at point \(B\) when \(t=a\), where \(a\) is a positive real constant.
- If the distance travelled along the curve from \(A\) to \(B\) is \(\dfrac{3 \pi}{4}\), find \(a\). (1 mark)
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- Find all values of \(t\) for which the position vector of the particle, \(\underset{\sim}{\text{r}}(t)\), is perpendicular to its velocity vector, \(\underset{\sim}{\dot{\text{r}}}(t)\). (2 marks)
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Functions, SPEC2 2023 VCAA 2 MC
The graph of \(y=\dfrac{x^3}{a x^2+b x+c}\) has asymptotes given by \(y=2 x+1\) and \(x=1\). The values of \(a, b\) and \(c\) are, respectively
- \(2,-4,2\)
- \( \dfrac{1}{2},-\dfrac{1}{4},-\dfrac{1}{4} \)
- \( \dfrac{1}{2}, \dfrac{1}{4},-\dfrac{3}{4} \)
- \( \dfrac{1}{2},-\dfrac{1}{4},-\dfrac{3}{4} \)
- \(2,-4,-8\)
Vectors, SPEC1 2023 VCAA 9
A plane contains the points \( A(1,3,-2), B(-1,-2,4)\) and \( C(a,-1,5)\), where \(a\) is a real constant. The plane has a \(y\)-axis intercept of 2 at the point \(D\).
- Write down the coordinates of point \(D\). (1 mark)
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- Show that \(\overrightarrow{A B}\) and \(\overrightarrow{A D}\) are \(-2 \underset{\sim}{\text{i}}-5 \underset{\sim}{\text{j}}+6 \underset{\sim}{\text{k}}\) and \(-\underset{\sim}{\text{i}}-\underset{\sim}{\text{j}}+2 \underset{\sim}{\text{k}}\), respectively. (1 mark)
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- Hence find the equation of the plane in Cartesian form. (2 marks)
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- Find \(a\). (1 mark)
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- \(\overline{A B}\) and \(\overline{A D}\) are adjacent sides of a parallelogram. Find the area of this parallelogram. (1 mark)
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Functions, SPEC1 2023 VCAA 1
Consider the function \(f\) with rule \(f(x)=\dfrac{x^2+x-6}{x-1}\).
- Show that the rule for the function \(f\) can be written as \(f(x)=x+2-\dfrac{4}{x-1}\). (1 mark)
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- Sketch the graph of \(f\) on the axes below, labelling any asymptotes with their equations. (3 marks)
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Statistics, SPEC1 2023 VCAA 6
Josie travels from home to work in the city. She drives a car to a train station, waits, and then rides on a train to the city. The time, \(X_c \) minutes, taken to drive to the station is normally distributed with a mean of 20 minutes \( (\mu_c=20) \) and standard deviation of 6 minutes \((\sigma_c=6) \). The waiting time, \( X_w \) minutes, for a train is normally distributed with a mean of 8 minutes \( (\mu_w=8) \) and standard deviation of \( \sqrt{3} \) minutes \( (\sigma_w=\sqrt{3}) \). The time, \( X_t \) minutes, taken to ride on a train to the city is also normally distributed with a mean of 12 minutes \( (\mu_t=12) \) and standard deviation of 5 minutes \( (\sigma_t=5) \). The three times are independent of each other.
- Find the mean and standard deviation of the total time, in minutes, it takes for Josie to travel from home to the city. (2 marks)
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- Josie's waiting time for a train on each work day is independent of her waiting time for a train on any other work day. The probability that, for 12 randomly chosen work days, Josie's average waiting time is between 7 minutes 45 seconds and 8 minutes 30 seconds is equivalent to \( \text{Pr}(a<Z<b)\), where \(Z \sim \text{N}(0,1)\) and \(a\) and \(b\) are real numbers.
- Find the values of \(a\) and \(b\). (2 marks)
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Calculus, SPEC1 2023 VCAA 4
Consider the relation \(x\, \arcsin \left(y^2\right)=\pi\).
Use implicit differentiation to find \(\dfrac{d y}{d x}\) at the point \(\left(6, \dfrac{1}{\sqrt{2}}\right)\).
Give your answer in the form \(-\dfrac{\pi \sqrt{a}}{b}\), where \(a, b \in Z^{+}\). (3 marks)
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Calculus, SPEC1 2023 VCAA 3
A particle moves along a straight line. When the particle is \(x\) m from a fixed point \(O\), its velocity, \( v\) m s\(^{-1}\), is given by
\(v=\dfrac{3 x+2}{2 x-1}\), where \(x \geq 1\).
- Find the acceleration of the particle, in m s\(^{-2}\), when \(x=2\). (2 marks)
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- Find the value that the velocity of the particle approaches as \(x\) becomes very large. (1 mark)
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Complex Numbers, SPEC1 2023 VCAA 2
Consider the complex number \(z=(b-i)^3\), where \(b \in R^{+}\).
Find \(b\) given that \(\arg (z)=-\dfrac{\pi}{2}\). (3 marks)
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Networks, GEN2 2023 VCAA 14
One of the landmarks in state \(A\) requires a renovation project.
This project involves 12 activities, \(A\) to \(L\). The directed network below shows these activities and their completion times, in days.
The table below shows the 12 activities that need to be completed for the renovation project.
It also shows the earliest start time (EST), the duration, and the immediate predecessors for the activities.
The immediate predecessor(s) for activity \(I\) and the EST for activity \(J\) are missing.
\begin{array} {|c|c|c|}
\hline
\quad \textbf{Activity} \quad & \quad\quad\textbf{EST} \quad\quad& \quad\textbf{Duration}\quad & \textbf{Immediate} \\
& & & \textbf{predecessor(s)} \\
\hline
\rule{0pt}{2.5ex} A \rule[-1ex]{0pt}{0pt} & 0 & 6 & - \\
\hline
\rule{0pt}{2.5ex} B \rule[-1ex]{0pt}{0pt} & 0 & 4 & - \\
\hline
\rule{0pt}{2.5ex} C \rule[-1ex]{0pt}{0pt} & 6 & 7 & A \\
\hline
\rule{0pt}{2.5ex} D \rule[-1ex]{0pt}{0pt} & 4 & 5 & B \\
\hline
\rule{0pt}{2.5ex} E \rule[-1ex]{0pt}{0pt} & 4 & 10 & B \\
\hline
\rule{0pt}{2.5ex} F \rule[-1ex]{0pt}{0pt} & 13 & 4 & C \\
\hline
\rule{0pt}{2.5ex} G \rule[-1ex]{0pt}{0pt} & 9 & 3 & D \\
\hline
\rule{0pt}{2.5ex} H \rule[-1ex]{0pt}{0pt} & 9 & 7 & D \\
\hline
\rule{0pt}{2.5ex} I \rule[-1ex]{0pt}{0pt} & 13 & 6 & - \\
\hline
\rule{0pt}{2.5ex} J \rule[-1ex]{0pt}{0pt} & - & 6 & E, H \\
\hline
\rule{0pt}{2.5ex} K \rule[-1ex]{0pt}{0pt} & 19 & 4 & F, I \\
\hline
\rule{0pt}{2.5ex} L \rule[-1ex]{0pt}{0pt} & 23 & 1 & J, K \\
\hline
\end{array}
- Write down the immediate predecessor(s) for activity \(I\). (1 mark)
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- What is the earliest start time, in days, for activity \(J\) ? (1 mark)
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- How many activities have a float time of zero? (1 mark)
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The managers of the project are able to reduce the time, in days, of six activities.
These reductions will result in an increase in the cost of completing the activity.
The maximum decrease in time of any activity is two days.
\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \textbf{Activity} \rule[-1ex]{0pt}{0pt} & \quad A \quad & \quad B \quad& \quad F \quad & \quad H \quad & \quad I \quad & \quad K \quad \\
\hline
\rule{0pt}{2.5ex} \textbf{Daily cost (\$)} \rule[-1ex]{0pt}{0pt} & 1500 & 2000 & 2500 & 1000 & 1500 & 3000 \\
\hline
\end{array}
- If activities \(A\) and \(B\) have their completion time reduced by two days each, the overall completion time of the project will be reduced.
- What will be the maximum reduction time, in days? (1 mark)
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- The managers of the project have a maximum budget of $15 000 to reduce the time for several activities to produce the maximum reduction in the project's overall completion time.
- Complete the table below, showing the reductions in individual activity completion times that would achieve the earliest completion time within the $ 15 000 budget. (1 mark)
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\begin{array} {|c|c|}
\hline
\quad\textbf{Activity} \quad & \textbf{Reduction in completion time} \\
& \textbf{(0, 1 or 2 days)}\\
\hline
\rule{0pt}{2.5ex} A \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} B \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} F \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} H \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} I \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} K \rule[-1ex]{0pt}{0pt} & \\
\hline
\end{array}
Networks, GEN2 2023 VCAA 13
The state \(A\) has nine landmarks, \(G, H, I, J, K, L, M, N\) and \(O\).
The edges on the graph represent the roads between the landmarks.
The numbers on each edge represent the length, in kilometres, along each road.
Three friends, Eden, Reynold and Shyla, meet at landmark \(G\).
- Eden would like to visit landmark \(M\).
- What is the minimum distance Eden could travel from \(G\) to \(M\) ? (1 mark)
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- Reynold would like to visit all the landmarks and return to \(G\).
- Write down a route that Reynold could follow to minimise the total distance travelled. (1 mark)
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- Shyla would like to travel along all the roads.
- To complete this journey in the minimum distance, she will travel along two roads twice.
- Shyla will leave from landmark \(G\) but end at a different landmark.
- Complete the following by filling in the boxes provided.
- The two roads that will be travelled along twice are the roads between: (1 mark)
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Networks, GEN2 2023 VCAA 12
A country has five states, \(A, B, C, D\) and \(E\).
A graph can be drawn with vertices to represent each of the states.
Edges represent a border shared between two states.
- What is the sum of the degrees of the vertices of the graph above? (1 mark)
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- Euler's formula, \(v+f=e+2\), holds for this graph.
- i. Complete the formula by writing the appropriate numbers in the boxes provided below. (1 mark)
- ii. Complete the sentence by writing the appropriate word in the space provided below. (1 mark)
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Euler’s formula holds for this graph because the graph is connected and ______________. - The diagram below shows the position of state \(A\) on a map of this country.
- The four other states are indicated on the diagram as 1, 2, 3 and 4.
- Use the information in the graph above to complete the table below. Match the state \((B, C, D\) and \(E)\) with the corresponding state number \((1,2,3\) and 4\()\) given in the map above. (1 mark)
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\hline
\rule{0pt}{2.5ex} \quad \quad \textbf{State} \quad \quad \rule[-1ex]{0pt}{0pt} & \textbf{State Number} \\
\hline
\rule{0pt}{2.5ex} B \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} C \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} D \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} E \rule[-1ex]{0pt}{0pt} & \\
\hline
\end{array}
Matrices, GEN2 2023 VCAA 10
Within the circus, there are different types of employees: directors \((D)\), managers \((M)\), performers \((P)\) and sales staff \((S).\) Customers \((C)\) attend the circus. Communication between the five groups depends on whether they are customers or employees, and on what type of employee they are. Matrix \(G\) below shows the communication links between the five groups. \begin{aligned} In this matrix: --- 2 WORK AREA LINES (style=lined) --- \begin{aligned} --- 0 WORK AREA LINES (style=lined) --- --- 2 WORK AREA LINES (style=lined) ---
&\quad \quad \quad\quad \quad \quad\quad \quad \quad \ \ \textit{receiver}\\
&\quad \quad\quad \quad \quad\quad \quad \quad D \ \ M \ \ P \ \ \ S \ \ \ C \\
& G=\textit{sender} \quad \begin{array}{ccccc}
D\\
M\\
P\\
S\\
C
\end{array}
\begin {bmatrix}
0 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 & 1 \\
0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0
\end{bmatrix}\\
&
\end{aligned}
&\quad \quad \quad\quad \quad \quad\quad \quad \quad \ \ \textit{receiver}\\
&\quad \quad\quad \quad \quad\quad \quad \quad D \quad M \quad P \quad \ S \quad \ C \\
& H=\textit{sender} \quad \begin{array}{ccccc}
D\\
M\\
P\\
S\\
C
\end{array}
\begin {bmatrix} {\displaystyle}
1 & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} \\
0 & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} \\
1 & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} \\
1 & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} \\
0 & 1 & 0 & 0 & 1
\end{bmatrix}\\
&
\end{aligned}
Matrices, GEN2 2023 VCAA 9
The circus is held at five different locations, \(E, F, G, H\) and \(I\).
The table below shows the total revenue for the ticket sales, rounded to the nearest hundred dollars, for the last 20 performances held at each of the five locations.
\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \textbf{Location} \rule[-1ex]{0pt}{0pt} & E & F & G & H & I \\
\hline
\rule{0pt}{2.5ex} \textbf{Ticket Sales} \rule[-1ex]{0pt}{0pt} & \$960\ 000 & \$990\ 500 & \$940\ 100 & \$920\ 800 & \$901\ 300 \\
\hline
\end{array}
The ticket sales information is presented in matrix \(R\) below.
\(R=\begin{bmatrix}
960\ 000 & 990\ 500 & 940\ 100 & 920\ 800 & 901\ 300
\end{bmatrix}\)
- Complete the matrix equation below that calculates the average ticket sales per performance at each of the five locations. (1 mark)
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\(\begin {bmatrix}\rule{2cm}{0.25mm} \end {bmatrix}\times R = \begin {bmatrix}\rule{2cm}{0.25mm} &\rule{2cm}{0.25mm} &\rule{2cm}{0.25mm} &\rule{2cm}{0.25mm} &\rule{2cm}{0.25mm} \end {bmatrix}\)
The circus would like to increase its total revenue from the ticket sales from all five locations.
The circus will use the following matrix calculation to target the next 20 performances.
\( [t] \times R \times \begin{bmatrix}
1 \\
1 \\
1 \\
1 \\
1
\end{bmatrix}\)
- Determine the value of \(t\) if the circus would like to increase its revenue from ticket sales by 25%. (1 mark)
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The circus moves from one location to the next each month. It rotates through each of the five locations, before starting the cycle again.
The following matrix displays the movement between the five locations.
\begin{aligned}
& \quad \ \ \ this \ month\\
& \ \ \ E \ \ \ F \ \ \ G \ \ \ H \ \ \ I \\
& \begin{bmatrix}
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0
\end{bmatrix} \begin{array}{ll}
E & \\
F\\
G & \ \ next \ month \\
H & \\
I
\end{array}\\
&
\end{aligned}
- The circus started in town \(I\).
- What is the order in which the circus will visit the five towns? (1 mark)
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Recursion and Finance, GEN2 2023 VCAA 6
Arthur invests $600 000 in an annuity that provides him with a monthly payment of $3973.00.
Interest is calculated monthly.
Three lines of the amortisation table for this annuity are shown below.
\begin{array} {|c|c|}
\hline
\textbf{Payment} & \textbf{Payment} & \textbf{Interest} & \textbf{Principal reduction} & \textbf{Balance} \\
\textbf{number} & \textbf{(\$) } & \textbf{(\$) } & \textbf{(\$) } & \textbf{(\$) }\\
\hline
\rule{0pt}{2.5ex} 0 \rule[-1ex]{0pt}{0pt} & 0.00 & 0.00 & 0.00 & 600\ 000.00 \\
\hline
\rule{0pt}{2.5ex} 1 \rule[-1ex]{0pt}{0pt} & 3973.00 & 2520.00 & 1453.00& 598\ 547.00\\
\hline
\rule{0pt}{2.5ex} 2 \rule[-1ex]{0pt}{0pt} & 3973.00 & 2513.90 & 1459.10 & 597\ 087.90 \\
\hline
\end{array}
- The interest rate for the annuity is 0.42% per month.
- Determine the interest rate per annum. (1 mark)
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- Using the values in the table, complete the next line of the amortisation table.
- Write your answers in the spaces provided in the table below.
- Round all values to the nearest cent. (1 mark)
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\begin{array} {|c|c|}
\hline
\textbf{Payment} & \textbf{Payment} & \textbf{Interest} & \textbf{Principal reduction} & \textbf{Balance} \\
\textbf{number} & \textbf{(\$) } & \textbf{(\$) } & \textbf{(\$) } & \textbf{(\$) }\\
\hline
\rule{0pt}{2.5ex} 0 \rule[-1ex]{0pt}{0pt} & 0.00 & 0.00 & 0.00 & 600\ 000.00 \\
\hline
\rule{0pt}{2.5ex} 1 \rule[-1ex]{0pt}{0pt} & 3973.00 & 2520.00 & 1453.00& 598\ 547.00\\
\hline
\rule{0pt}{2.5ex} 2 \rule[-1ex]{0pt}{0pt} & 3973.00 & 2513.90 & 1459.10 & 597\ 087.90 \\
\hline
\rule{0pt}{2.5ex} 3 \rule[-1ex]{0pt}{0pt} & & & & \\
\hline
\end{array}
- Let \(V_n\) be the balance of Arthur's annuity, in dollars, after \(n\) months.
- Write a recurrence relation in terms of \(V_0, V_{n+1}\) and \(V_n\) that can model the value of the annuity from month to month. (1 mark)
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- The amortisation tables above show that the balance of the annuity reduces each month.
- If the balance of an annuity remained constant from month to month, what name would be given to this type of annuity? (1 mark)
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Recursion and Finance, GEN2 2023 VCAA 5
Arthur borrowed $30 000 to buy a new motorcycle. Interest on this loan is charged at the rate of 6.4% per annum, compounding quarterly. Arthur will repay the loan in full with quarterly repayments over six years. --- 1 WORK AREA LINES (style=lined) --- The balance of the loan, in dollars, after \(n\) quarters, \(A_n\), can be modelled by the recurrence relation \(A_0=30\ 000, \quad A_{n+1}=1.016 A_n-1515.18\) --- 4 WORK AREA LINES (style=lined) --- --- 3 WORK AREA LINES (style=lined) ---
Data Analysis, GEN2 2023 VCAA 4
The time series plot below shows the average monthly ice cream consumption recorded over three years, from January 2010 to December 2012.
The data for the graph was recorded in the Northern Hemisphere.
In this graph, month number 1 is January 2010, month number 2 is February 2010 and so on.
- Identify a feature of this plot that is consistent with this time series having a seasonal component. (1 mark)
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- The long-term seasonal index for April is 1.05
- Determine the deseasonalised value for average monthly ice cream consumption in April 2010 (month 4).
- Round your answer to two decimal places. (1 mark)
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- The table below shows the average monthly ice cream consumption for 2011 .
Consumption (litres/person) | ||||||||||||
Year | Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sept | Oct | Nov | Dec |
2011 | 0.156 | 0.150 | 0.158 | 0.180 | 0.200 | 0.210 | 0.183 | 0.172 | 0.162 | 0.145 | 0.134 | 0.154 |
- Show that, when rounded to two decimal places, the seasonal index for July 2011 estimated from this data is 1.10. (2 marks)
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Data Analysis, GEN2 2023 VCAA 3
The scatterplot below plots the average monthly ice cream consumption, in litres/person, against average monthly temperature, in °C. The data for the graph was recorded in the Northern Hemisphere.
When a least squares line is fitted to the scatterplot, the equation is found to be:
consumption = 0.1404 + 0.0024 × temperature
The coefficient of determination is 0.7212
- Draw the least squares line on the scatterplot graph above. (1 mark)
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- Determine the value of the correlation coefficient \(r\).
- Round your answer to three decimal places. (1 mark)
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- Describe the association between average monthly ice cream consumption and average monthly temperature in terms of strength, direction and form. (1 mark)
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\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \textbf{strength} \rule[-1ex]{0pt}{0pt} & \quad \quad \quad \quad \quad \quad \quad \quad \\
\hline
\rule{0pt}{2.5ex} \textbf{direction} \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} \textbf{form} \rule[-1ex]{0pt}{0pt} & \\
\hline
\end{array} - Referring to the equation of the least squares line, interpret the value of the intercept in terms of the variables consumption and temperature. (1 mark)
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- Use the equation of the least squares line to predict the average monthly ice cream consumption, in litres per person, when the monthly average temperature is –6°C. (1 mark)
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- Write down whether this prediction is an interpolation or an extrapolation. (1 mark)
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Proof, EXT2 P2 2023 SPEC1 8
A function \(f\) has the rule \(f(x)=x\,e^{2x}\).
Use mathematical induction to prove that \(f^{(n)}(x)=\big{(}2^{n}x+n\,2^{n-1}\big{)}e^{2x}\) for \(n \in \mathbb{Z}^{+}\), where \(f^{(n)}(x)\) represents the \(n\)th derivative of \(f(x)\). That is, \(f(x)\) has been differentiated \(n\) times. (3 marks)
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Induction, SPEC1 2023 VCAA 8
A function \(f\) has the rule \(f(x)=x\,e^{2x}\).
Use mathematical induction to prove that \(f^{(n)}(x)=\big{(}2^{n}x+n\,2^{n-1}\big{)}e^{2x}\) for \(n \in \mathbb{Z}^{+}\), where \(f^{(n)}(x)\) represents the \(n\)th derivative of \(f(x)\). That is, \(f(x)\) has been differentiated \(n\) times. (4 marks)
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Matrices, GEN1 2022 VCAA 5 MC
Matrix \(E\) is a 2 × 2 matrix.
Matrix \(F\) is a 2 × 3 matrix.
Matrix \(G\) is a 3 × 2 matrix.
Matrix \(H\) is a 3 × 3 matrix.
Which one of the following matrix products could have an inverse?
- \(EF\)
- \(FH\)
- \(GE\)
- \(GF\)
- \(HG\)
Matrices, GEN1 2022 VCAA 3 MC
Each day, members of a swim centre can choose to attend a morning session \((M)\), an afternoon session \((A)\) or no session \((N)\).
The transition diagram below shows the transition from day to day.
The transition diagram is incomplete.
Which one of the following transition matrices represents this transition diagram?
Matrices, GEN1 2022 VCAA 1-2 MC
A bike rental business rents road bikes \((R)\) and mountain bikes \((M)\) in three sizes: child \((C)\), junior \((J)\) and adult \((A)\).
Matrix \(B\) shows the daily rental cost, in dollars, for each type of bike.
\begin{aligned} \\
& \quad R \ \quad \ \ M \\
B = & \begin{bmatrix}
80 & 95 \\
110 & 120 \\
120 & 125
\end{bmatrix}\begin{array}{l}
C\\
J\\
A
\end{array}
\end{aligned}
The element in row \(i\) and column \(j\) in matrix \(B\) is \(b_{ij}\).
Question 1
The daily cost of renting an adult mountain bike is shown in element
- \(b_{12}\)
- \(b_{21}\)
- \(b_{23}\)
- \(b_{31}\)
- \(b_{32}\)
Question 2
On Sundays, the business increases the daily rental price for each type of bike by 10%.
To determine the rental cost for each type of bike on a Sunday, which one of the following matrix calculations needs to be completed?
- \(0.01B\)
- \(0.1B\)
- \(1.01B\)
- \(1.1B\)
- \(11B\)
Recursion and Finance, GEN1 2022 VCAA 22 MC
Tim deposited $6000 into an investment account earning compound interest calculated monthly.
A rule for the balance, \(T_n\), in dollars, after \(n\) years is given by \(T_n=6000 \times 1.003^{12n}\).
Let \(R_n\) be a new recurrence relation that models the balance of Tim's account after \(n\) months.
This recurrence relation is
- \(R_0=6000,\ \ \ R_{n+1}=R_n+18\)
- \(R_0=6000,\ \ \ R_{n+1}=R_n+36\)
- \(R_0=6000,\ \ \ R_{n+1}=1.003\,R_n\)
- \(R_0=6000,\ \ \ R_{n+1}=1.0036\,R_n\)
- \(R_0=6000,\ \ \ R_{n+1}=1.036\,R_n\)
Recursion and Finance, GEN1 2022 VCAA 20 MC
Nidhi owns equipment that is used for 10 hours per day for all 365 days of the year.
The value of the equipment is depreciated by Nidhi using the unit cost method.
The value of the equipment, \(E_n\), in dollars, after \(n\) years can be modelled by the recurrence relation
\(E_0=100\ 000,\ \ \ E_{n+1}=E_n-5475\)
The value of the equipment is depreciated by
- $1.50 per hour.
- $10 per hour.
- $15 per hour.
- $1.50 per day.
- $10 per day.
Data Analysis, GEN1 2022 VCAA 15 MC
The daily number of cups of coffee sold by a food truck over a three-week period is shown in the table below.
The six-mean smoothed number of cups of coffee, with centring, sold on Thursday in Week 2 is closest to
- 127
- 138
- 147
- 155
- 163
Data Analysis, GEN1 2022 VCAA 12-14 MC
The scatterplot below displays the body length, in centimetres, of 17 crocodiles, plotted against their head length, in centimetres. A least squares line has been fitted to the scatterplot. The explanatory variable is head length.
Question 12
The equation of the least squares line is closest to
- head length = –40 + 7 × body length
- body length = –40 + 7 × head length
- head length = 168 + 7 × body length
- body length = 168 – 40 × head length
- body length = 7 + 168 × head length
Question 13
The median head length of the 17 crocodiles, in centimetres, is closest to
- 49
- 51
- 54
- 300
- 345
Question 14
The correlation coefficient \(r\) is equal to 0.963
The percentage of variation in body length that is not explained by the variation in head length is closest to
- 0.9%
- 3.7%
- 7.3%
- 92.7%
- 96.3%
CHEMISTRY, M7 2020 VCE 3
Below is a reaction pathway beginning with hex-3-ene.
- Write the IUPAC name of Compound J in the box provided. (1 mark)
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- State the reagent(s) required to convert hex-3-ene to hexan-3-ol in the box provided. (1 mark)
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- Draw the structural formula for a tertiary alcohol that is an isomer of hexan-3-ol. (1 mark)
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- Hexan-3-ol is reacted with Compound M under acidic conditions to produce Compound L.
- Draw the semi-structural formula for Compound M in the box provided on the image above. (1 mark)
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- i. Draw the semi-structural formula for Compound K in the box provided on the image above. (1 mark)
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- ii. Name the class of organic compound (homologous series) to which Compound K belongs. (1 mark)
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- What type of reaction produces Compound K from hexan-3-ol? (1 mark)
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